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LIST EDGE AND LIST TOTAL COLORINGS OF PLANAR GRAPHS WITHOUT 6-CYCLES WITH CHORD

2012
*
Bulletin of the Korean Mathematical Society
*

It is proved that if a

doi:10.4134/bkms.2012.49.2.359
fatcat:oxtudhudpndhpd2d6sk7ffgoea
*planar**graph*G*without**6*-*cycles**with**chord*, then χ ′ l ... Giving a*planar**graph*G, let χ ′ l (G) and χ ′′ l (G) denote the*list**edge*chromatic number and*list*total chromatic number*of*G respectively. ...*Planar**graphs**without**6*-*cycles**with**chord*First let us introduce an important lemma. Proof. For G is a critical*planar**graph**with*∆(G) ≥*6*, then G contains no (4, 4, 5 − )-face f = uvw. ...##
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3-Choosability of Triangle-Free Planar Graphs with Constraints on 4-Cycles

2010
*
SIAM Journal on Discrete Mathematics
*

A

doi:10.1137/080743020
fatcat:qq6666y2b5bk5f5kw3reox6rbu
*graph*is k-choosable if it can be*colored*whenever every vertex has a*list**of*at least k available*colors*. A theorem by Grötzsch [2] asserts that every triangle-free*planar**graph*is 3-*colorable*. ... In addition, this implies that every triangle-free*planar**graph**without**6*-and 7-*cycles*is 3-choosable. * Supported by a CZ-SL bilateral project MEB 090805 and BI-CZ/08-09-005. ... This strengthens the results*of*Lidický [*6*] that*planar**graphs**without*3-,*6*-, 7-and 8-*cycles*are 3-choosable, and*of*Zhang and Xu [13] that*planar**graphs**without*3-,*6*-, 7-and 9-*cycles*are 3-choosable ...##
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Correspondence coloring and its application to list-coloring planar graphs without cycles of lengths 4 to 8
[article]

2016
*
arXiv
*
pre-print

Using this tool, we prove that excluding

arXiv:1508.03437v2
fatcat:iqjnicgb5vcthd2r3flfhifai4
*cycles**of*lengths 4 to 8 is sufficient to guarantee 3-choosability*of*a*planar**graph*, thus answering a question*of*Borodin. ... We introduce a new variant*of**graph**coloring*called correspondence*coloring*which generalizes*list**coloring*and allows for reductions previously only possible for ordinary*coloring*. ... On the other hand,*planar**graphs*are 5-choosable [14] , and every*planar**graph**without**cycles**of*lengths 3 and 4 is 3-choosable [15] . ...##
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List Edge Colorings of Planar Graphs with 7-cycles Containing at Most Two Chords

2019
*
Applied Science and Innovative Research
*

In this paper we prove that if G is a

doi:10.22158/asir.v3n2p85
fatcat:n5lr2zt5ungkri6v7de2edl6em
*planar**graph*, and each 7-*cycle*contains at most two*chords*, then G is*edge*-k-choosable, where k = max{8, ?(G) + 1}. ... Structural Properties*of**Planar**Graphs**with*7-*Cycles*Containing at Most Two*Chords*Lemma 4. Let G be a*planar**graph*in which each 7-*cycle*contains at most two*chords*. ... Suppose that G is a counterexample to our theorem*with*the minimum number*of**edges*and G is any*planar**graph*in which every 7-*cycle*contains at most two*chords*. ...##
###
On oriented cliques with respect to push operation
[article]

2015
*
arXiv
*
pre-print

To push a vertex v

arXiv:1511.08672v1
fatcat:gayvb57bpngojgdr77mazhrqra
*of*a directed*graph*G is to change the orientations*of*all the arcs incident*with*v. An oriented*graph*is a directed*graph**without*any*cycle**of*length at most 2. ... We also provide an exhaustive*list**of*minimal (*with*respect to spanning subgraph inclusion)*planar*push cliques. ... Note that an eight*cycle*can have*three*types*of**chords*, namely, a very long*chord*connecting vertices at distance 4, a long*chord*and a short*chord*. ...##
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On Choosability with Separation of Planar Graphs with Forbidden Cycles

2015
*
Journal of Graph Theory
*

We prove that

doi:10.1002/jgt.21875
fatcat:5z7apx6gsredpmlpgdu2gujdn4
*planar**graphs**without*4-*cycles*are (3,1)-choosable and that*planar**graphs**without*5-*cycles*and*6*-*cycles*are (3,1)-choosable. ... We study choosability*with*separation which is a constrained version*of**list**coloring**of**graphs*. ... Jao for fruitful discussions and encouragement in the early stage*of*the project. ...##
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Algorithms
[chapter]

2011
*
Graph Coloring Problems
*

*Graphs*97 5.1 Critical

*Graphs*

*With*Many

*Edges*97 5.2 Minimum Degree

*of*4-and 5-Critical

*Graphs*> 98 5.3 Critical

*Graphs*

*With*Few

*Edges*99 5.4 Four-Critical Amenable

*Graphs*101 5.5 Four-Critical ...

*of*Hamilton

*Cycles*. 82 4.6 Brooks' Theorem for Triangle-Free

*Graphs*83 4.7

*Graphs*

*Without*Large Complete Subgraphs 85 4.8 ^-Chromatic

*Graphs*

*of*Maximum Degree k 85 4.9 Total

*Coloring*86 ...

##
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Do triangle-free planar graphs have exponentially many 3-colorings?
[article]

2017
*
arXiv
*
pre-print

Thomassen conjectured that triangle-free

arXiv:1702.00588v2
fatcat:xv4etv6fkfaorkxvpkzg2qdv4q
*planar**graphs*have an exponential number*of*3-*colorings*. ... We show this conjecture to be equivalent to the following statement: there exists a positive real α such that whenever G is a*planar**graph*and A is a subset*of*its*edges*whose deletion makes G triangle-free ...*coloring**of**planar**graphs*. ...##
###
Planar graphs without cycles of length 4, 7, 8, or 9 are 3-choosable

2011
*
Discrete Applied Mathematics
*

It is known that

doi:10.1016/j.dam.2010.11.002
fatcat:g6hk5nmh6rhjnaefrrxjcnwmlm
*planar**graphs**without**cycles**of*length 4, i, j, or 9*with*4 < i < j < 9, except that i = 7 and j = 8, are 3-choosable. ... This paper proves that*planar**graphs**without**cycles**of*length 4, 7, 8, or 9 are also 3-choosable. ... Acknowledgements The authors are grateful to the referees for their wise suggestions to simplify the original proof*of*this paper. ...##
###
Improper choosability of graphs of nonnegative characteristic

2008
*
Computers and Mathematics with Applications
*

In this paper, we prove: (1) If G contains no k-

doi:10.1016/j.camwa.2008.03.036
fatcat:437w5yexyrdplp74g6fmfvu3g4
*cycle**with*a*chord*for all k = 4, 5,*6*, then G is (3, 1) *choosable; (2) If G contains neither 5-*cycle**with*a*chord*nor*6*-*cycle**with*a*chord*, then G is ( ... A*graph*G is called (k, d) * -choosable if, for every*list*assignment L*with*|L(v)| = k for all v ∈ V (G), there is an L-*coloring**of*G such that every vertex has at most d neighbors having the same*color*... Equivalently, every torodial*graph**without*a 4-*cycle**with*a*chord*is (4, 1) * -choosable. Theorems 5 and*6*are, to some extent, an extension to their result. ...##
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List coloring of planar graphs without 6-cycles with two chords

2018
*
Discussiones Mathematicae Graph Theory
*

If ∆(G) ≥

doi:10.7151/dmgt.2183
fatcat:le3o6s6lzna35ebvqzdgudd57i
*6*, then χ ′*list*(G) ≤ ∆(G) + 1. The following result is about*edge*-∆-choosable*of*embedded*planar**graphs**without**6*-*cycles**with*two*chords*. Theorem 8. ... Main Results and Their Proofs In the section, we always assume that all*graphs*are*planar**graphs*that have been embedded in the plane and G is a*planar**graph**without**6*-*cycles**with*two*chords*. ... By Lemmas 9 and 10, v may be the 3master*of*two 3 − -vertices and the 2-master*of*a 2-vertex, that is, v sends at most 5 to its incident 3 − -vertices. Suppose that d(v) = 9. Then f 3 (v) ≤*6*. ...##
###
(4, 2)-Choosability of Planar Graphs with Forbidden Structures

2017
*
Graphs and Combinatorics
*

A

doi:10.1007/s00373-017-1812-5
fatcat:ta5azl3h7nhhfoqnlgnmjmyswm
*chorded*ℓ-*cycle*is an ℓ-*cycle**with*one additional*edge*. We demonstrate for each ℓ∈{5,6,7} that a*planar**graph*is (4,2)-choosable if it does not contain*chorded*ℓ-*cycles*. ... A*graph*is (k,s)-choosable if the*graph*is*colorable*from*lists**of*size k where adjacent vertices have at most s common*colors*in their*lists*. ... Conclusion and Future Work We proved that, for each k ∈ {5,*6*, 7},*planar**graphs**with*no*chorded*k-*cycles*are (4, 2)-choosable. ...##
###
On oriented cliques with respect to push operation

2017
*
Discrete Applied Mathematics
*

An oriented

doi:10.1016/j.dam.2017.07.037
fatcat:na4wiolki5gp3eq2653zyy73ne
*graph*is a directed*graph**without*any directed*cycle**of*length at most 2. An oriented clique is an oriented*graph*whose non-adjacent vertices are connected by a directed 2-path. ... We also prove that a*planar*push clique can have at most 8 vertices and provide an exhaustive*list**of**planar*push cliques. ... Acknowledgement: The authors would like to thank the anonymous reviewer for the constructive comments towards improvement*of*the content, clarity and conciseness*of*the manuscript. ...##
###
Planar graphs without normally adjacent short cycles
[article]

2021
*
arXiv
*
pre-print

Consequently, every

arXiv:1908.04902v3
fatcat:fho4mz3b5vdpldbq2eobl6jgqi
*planar**graph**without*4-,*6*-, 8-*cycles*is 3-choosable, and every*planar**graph**without*4-, 5-, 7-, 8-*cycles*is 3-choosable. ... Let 𝒢 be the class*of*plane*graphs**without*triangles normally adjacent to 8^--*cycles*,*without*4-*cycles*normally adjacent to*6*^--*cycles*, and*without*normally adjacent 5-*cycles*. ... This work was supported by the National Natural Science Foundation*of*China and partially supported by the Fundamental Research Funds for Universities in Henan (YQPY20140051). ...##
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On choosability with separation of planar graphs with lists of different sizes

2015
*
Discrete Mathematics
*

A

doi:10.1016/j.disc.2015.01.008
fatcat:njxw7lf4j5fijlqswvwnwg6xea
*graph*G is (k,d)-choosable if there exists an L-*coloring**of*G for every (k,d)-*list*assignment L. This concept is also known as choosability*with*separation. ... A (k,d)-*list*assignment L*of*a*graph*G is a mapping that assigns to each vertex v a*list*L(v)*of*at least k*colors*and for any adjacent pair xy, the*lists*L(x) and L(y) share at most d*colors*. ... Choi et. al [1] proved that every*planar**graph**without*4-*cycles*is (3, 1)-choosable and that every*planar**graph**without*5-*cycles*and*6*-*cycles*is (3, 1)-choosable. ...
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